Weighted Chairman Assignment and Flow-Time Scheduling (2511.18546v1)

Abstract: Given positive integers $m, n$, a fractional assignment $x \in [0,1]^{m \times n}$ and weights $d \in \mathbb{R}^n_{>0}$, we show that there exists an assignment $y \in {0,1}^{m \times n}$ so that for every $i\in[m]$ and $t\in [n]$, [ \Big|\sum_{j \in [t]} d_j (x_{ij} - y_{ij}) \Big| < \max_{j \in [n]} d_j. ] This generalizes a result of Tijdeman (1973) on the unweighted version, known as the chairman assignment problem. This also confirms a special case of the single-source unsplittable flow conjecture with arc-wise lower and upper bounds due to Morell and Skutella (IPCO 2020). As an application, we consider a scheduling problem where jobs have release times and machines have closing times, and a job can only be scheduled on a machine if it is released before the machine closes. We give a $3$-approximation algorithm for maximum flow-time minimization.

• The paper introduces a deterministic algorithm that rounds fractional assignments with a weighted discrepancy at most (1 - 1/(2m-2)) times the maximum weight.
• It develops combinatorial techniques and LP rounding to yield a (3 - 1/(m-1))-approximation algorithm for flow-time scheduling under machine closing constraints.
• The work establishes tight lower bounds and connects weighted chairman assignment with the unsplittable flow conjecture, significantly impacting scheduling and network flow theory.

Weighted Chairman Assignment and Flow-Time Scheduling

Overview

The paper "Weighted Chairman Assignment and Flow-Time Scheduling" (2511.18546) presents a significant generalization of the classic chairman assignment problem. The authors introduce a weighted version of the problem, resolve a core open case related to the single-source unsplittable flow conjecture, and provide novel approximation algorithms for flow-time scheduling problems involving release times and machine closing constraints. The work advances discrepancy minimization and combinatorial rounding, with implications for scheduling theory, network flows, and combinatorial discrepancy theory.

The Weighted Chairman Assignment Problem

The classical chairman assignment problem investigates how to round a fractional assignment $x \in [0, 1]^{m \times n}$, where each column sums to one, to an integral assignment $y \in \{0,1\}^{m \times n}$, such that prefix sums in every row track their fractional counterparts up to small additive error. The discrepancy, i.e., the maximal difference across all rows and prefixes, is historically tightly bounded in the unweighted ($d_j=1$) case.

This work generalizes to the weighted setting: each column $j$ is associated with an arbitrary positive weight $d_j$, and the objective is to construct $y$ so $|\sum_{j=1}^t d_j (x_{ij} - y_{ij})|$ is minimized for all rows $i$ and prefixes $t$. Extant structural and combinatorial arguments in the unweighted case (Hall’s theorem, bipartite matchings, etc.) do not transfer due to the loss of integrality in updates.

The main result is a deterministic, linear-time algorithm that, given any $x$ and $d \in \mathbb{R}^n_{>0}$, constructs $y$ such that the weighted discrepancy is at most $(1-\frac{1}{2m-2})\max_j d_j$ across all rows and prefixes. This analysis recovers tight bounds in the unweighted setting and achieves the best possible up to the lower bound constructions.

Relation to the Unsplittable Flow Conjecture

A core combinatorial abstraction underlying this work is the single-source unsplittable flow problem with arc-wise lower and upper bounds. Prior conjectures (Morell and Skutella; see (2511.18546)) posit that for any fractional flow satisfying terminal demands $d_1,\ldots,d_n$, there is an unsplittable flow (i.e., paths for each demand) matching each arc up to the largest single demand. The weighted chairman assignment problem is equivalent to a special (path-structured) instance of this conjecture.

This work fully resolves this case, constructing assignments with prefix discrepancy at most $\max_j d_j$, which was previously known only for uniform and highly structured demand vectors. All prior linear programming and probabilistic discrepancy techniques break down in the presence of nonuniform weights, so this combinatorial approach represents a rigorous advance.

Lower Bounds and Tightness

The paper complements algorithmic results with new, more transparent lower-bound constructions. These match the upper bounds (up to constant additive factors), demonstrating tightness of the main result. Specifically, for any $m>1$, there exists a fractional assignment such that every integer rounding must incur weighted discrepancy at least $1-\frac{1}{2m-2}$, even in the unweighted setting.

Moreover, the influence of constraining the support (columns with nonzero fractional assignment per row) is investigated, and the authors refute a conjecture that optimal prefix discrepancy could be guaranteed over all sub-intervals (not just prefixes) — providing a counterexample with substantial excess.

Applications to Flow-Time Scheduling

As an important application, the authors consider maximum flow-time minimization problems in scheduling under eligibility and time window constraints. Given jobs with release times, processing times, and machine closing times, the task is to minimize the maximum flow-time (completion minus release).

By combining the weighted rounding procedure with suitable assignment and duality gadgets in the LP relaxation, the authors design a $(3-\frac{1}{m-1})$-approximation algorithm. This strictly improves upon best known previous ratios for non-identical closing times and shows that strong discrepancy rounding bounds can directly yield near-optimal approximation guarantees for complex scheduling objectives.

It is shown that the commonly used greedy algorithm, FIFO, has an approximation lower bound of $\Omega(\log m)$ in these settings, highlighting the superiority of the approach. In addition, they confirm a central conjecture in the literature for the special case of machine closing times and polish the connection between combinatorial discrepancy and scheduling integrality gaps.

Structural and Theoretical Implications

The results have far-reaching implications for several active areas:

1. Discrepancy Theory: The weighted generalization opens up technical pathways for extending classic rounding theorems e.g., Beck-Fiala-type results, to weighted and prefix-constrained discrepancy minimization.
2. Network Flows: The work leverages and complements results on unsplittable flows in acyclic digraphs, particularly path and series-parallel structures, where powerful combinatorial and probabilistic approaches have previously been stymied by weights.
3. Scheduling: The translation of combinatorial discrepancy bounds to scheduling guarantees demonstrates how close integrality of job assignments can be achieved in non-trivial machine eligibility and time-windowed settings. The results suggest that $(1+o(1))$-approximate solutions may be achievable in broader variants, contingent on refined discrepancy techniques.
4. Open Problems: The study articulates challenging open conjectures, including non-uniform per-machine/per-job weights, and assignment under additional structural constraints (e.g., carpooling/capacity, committee assignments). Lower-bound constructions indicate subtle limitations on achievable discrepancy, especially outside the prefix regime.

Conclusion

This paper delivers a comprehensive treatment of the weighted chairman assignment problem, providing optimal deterministic algorithms, strong lower bounds, and direct algorithmic applications to flow-time scheduling with eligibility and time windows. The connection to unsplittable flow rounding is firmly established, and the techniques developed are robust enough to suggest fruitful future advances in discrepancy minimization and its broad applications in scheduling and combinatorial optimization.